Damping From Log Decrement⚠ unverified

Understanding the Parameters

• Logarithmic decrement $\delta$ — the natural log of the ratio of two successive peak amplitudes, $\delta = \ln(x_1/x_2)$. Lightly damped systems decay slowly ($\delta$ small); heavily damped ones decay fast ($\delta$ large). It is measured directly from a ring-down (free-decay) test.

• Exact vs approximate — for light damping the relation simplifies to $\zeta \approx \delta/2\pi$, since the $\delta^2$ term under the root is negligible. The full form here is exact for all $\zeta < 1$ and matters as damping grows.

• Why measure it this way — a free-decay test needs only a displacement/acceleration sensor and a disturbance (a tap or release); no force measurement or frequency sweep is required, making log decrement the cheapest, most robust damping measurement.

• Averaging over cycles — in practice $\delta$ is taken over $n$ cycles, $\delta = \tfrac1n\ln(x_0/x_n)$, to reduce noise before applying this inversion.

Derivation (Approaching a Proof)

An underdamped SDOF system decays as $x(t) = X e^{-\zeta\omega_n t}\cos(\omega_d t + \phi)$, with damped frequency $\omega_d = \omega_n\sqrt{1-\zeta^2}$ and period $T_d = 2\pi/\omega_d$. The ratio of amplitudes one period apart is the decay of the exponential envelope over $T_d$:

$$\frac{x_1}{x_2} = e^{\zeta\omega_n T_d} \;\Longrightarrow\; \delta = \ln\frac{x_1}{x_2} = \zeta\omega_n T_d = \frac{2\pi\zeta\omega_n}{\omega_d} = \frac{2\pi\zeta}{\sqrt{1-\zeta^2}}.$$

Now solve this for $\zeta$. Squaring, $\delta^2(1-\zeta^2) = 4\pi^2\zeta^2$, so $\delta^2 = \zeta^2(4\pi^2 + \delta^2)$, giving

$$\zeta = \frac{\delta}{\sqrt{4\pi^2 + \delta^2}} = \frac{\delta}{\sqrt{(2\pi)^2 + \delta^2}}.$$

For $\zeta \ll 1$ the $\delta^2$ term drops and $\zeta \approx \delta/2\pi$.

Dimensional check. $\delta$ and $2\pi$ are dimensionless, so $\zeta = \delta/\sqrt{(2\pi)^2+\delta^2}$ is dimensionless — a ratio, as required.

History and Development

The logarithmic-decrement method is one of the oldest experimental damping measurements, dating to the 19th-century study of decaying oscillations. Its exact inversion to $\zeta$ is standard in every vibration text (Thomson, Rao) and remains the go-to technique for measuring modal damping from an impulse ("bump" or "pluck") test in the field, complementing the frequency-domain half-power-bandwidth method (Half Power Bandwidth).

Related Concepts: Logarithmic Decrement, Damping Ratio, Half Power Bandwidth, Quality Factor Q, Damped Natural Frequency, Critical Damping

Notes: Exact inversion of $\delta = 2\pi\zeta/\sqrt{1-\zeta^2}$. Light damping → $\zeta \approx \delta/2\pi$. Measured from a free-decay (ring-down) test; average over $n$ cycles for accuracy. Complements the frequency-domain Half Power Bandwidth method.