Logarithmic Decrement

Damping is the one parameter in vibration analysis that cannot be calculated from a drawing. Mass comes off the weights, stiffness off the geometry and the modulus, but damping depends on joints, welds, coatings, bolt tension and how the machine happens to be sitting — so it has to be measured. The logarithmic decrement is how: hit the structure, let it ring down, and read the decay off the trace. $\delta = \frac{1}{n}\ln(x_1/x_2)$.

The reason a logarithm appears is that a damped free oscillation decays exponentially, $x(t) = X e^{-\zeta\omega_n t}\cos(\omega_d t + \phi)$. Successive peaks are one damped period apart, so their ratio is the same constant every cycle — the third peak is the same fraction of the second as the second is of the first. Taking the natural log of that constant ratio turns a multiplicative decay into a number that simply adds up cycle by cycle, which is why dividing by $n$ works and why averaging over many cycles is legitimate.

The result you actually want is the damping ratio, and it comes from $\zeta = \delta/\sqrt{4\pi^2 + \delta^2}$. For light damping this simplifies to $\zeta \approx \delta/2\pi$, which is within 1% up to about $\zeta = 0.2$ and is the form usually quoted. A decrement of 0.1 means a damping ratio of about 0.016 — and it also means each swing is about 90% of the one before, which is a decay slow enough that a bystander would call the machine "ringy".

Three things spoil the measurement. Use several cycles rather than two adjacent peaks: any single peak carries noise, and $n = 5$ or 10 averages it out. Compare peaks on the SAME side of the zero line, because a trace riding on a DC offset gives different answers for its two halves. And make sure it really is a free decay — the machine off, nearby equipment off, no forcing of any kind — since anything still driving the structure contaminates the record. One more reading is worth knowing: if the amplitude GROWS during a free decay, that is not a sign error. It is self-excitation — oil whirl, stick-slip, aerodynamic flutter — and it is a far more serious problem than the damping measurement it interrupted.