A typical free vibrational displacement vs. time data set is show to the right. Recall from the Free Vib Eqn Solution that the solution to the differential equation for free response is:
(1)
Notice that the response remains within an exponential envelope given by:
(2)
with a time constant given by:
(3)
By determining the damped natural frequency -- or ringing frequency -- and the time constant of the exponential envelope, the natural frequency and the damping ratio can be determined.