the logarithmic decrement
/ log-DEK-ruh-ment /
Suppose you watch a damped vibration ring down and want to measure how quickly it loses energy — but you cannot see inside the system to read off c, m, and k. The logarithmic decrement is a clever trick: just compare the heights of two successive peaks of the wobble. Their ratio tells you the damping, no disassembly required. It is how you reverse-engineer the damping of a real, sealed device from the outside.
For an underdamped system, each peak is smaller than the one before by a fixed factor, because the decaying envelope e^(-(c/2m) t) multiplies the motion. The logarithmic decrement is defined as delta = ln(x_n / x_(n+1)), the natural log of the ratio of one peak to the next. Working through the envelope shows this equals delta = 2 pi zeta / sqrt(1 - zeta^2), a direct, clean link to the damping ratio zeta. Measure two peaks, take the log of their ratio, and you can solve back for zeta.
This makes the logarithmic decrement a favourite experimental tool: tap a structure, record the ring-down on a sensor, read two peak heights, and you have the damping of a bridge cable, a turbine blade, or a building — all without knowing its mass or stiffness. For light damping the formula simplifies to delta ≈ 2 pi zeta, so each peak is roughly (1 - 2 pi zeta) times the one before, a quick mental estimate.
If one peak is 8 mm and the next is 5 mm, then delta = ln(8/5) ≈ 0.470; solving 0.470 = 2 pi zeta / sqrt(1 - zeta^2) gives zeta ≈ 0.074 — a lightly damped system.
Two peak heights and one logarithm reveal the hidden damping ratio.
The logarithmic decrement only makes sense for underdamped motion — a critically or overdamped system has no successive peaks to compare. For more accuracy on a very lightly damped system, average over n cycles: delta = (1/n) ln(x_0 / x_n).