It's important that the seismometer be sufficiently
damped. The "damping ratio" is the ratio of the actual
damping to critical damping. If the seismometer is critically
damped, the damping ratio is 1.0. A damping ratio of 0.7 to 0.8 is
best. To determine the damping ratio the amplitude of succeeding
peaks is measured, as in the diagram to the left.
In the table below, enter one peak as X and the next as Y. For example enter X1 and X2 or enter X2 and X3. The sign of both numbers should be positive. Then press the Damping Ratio button. |

by The JavaScript Source

The equations:

Let the Damping Ratio be z and the amplitude of two successive peaks be A1 and A2.

ln (A1 / A2) = Pi * z / sqrt ( 1 - z**2)

z = 1 / sqrt {1 + [Pi / ln(A1 / A2)]**2}

See also W. Bauer's applet for an animation of a damped oscillator.