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Since I was going to get the damping ratio from time domain, the first thought was using logarithmic decrement method. But I'm not sure if I can use it when the given data is acceleration-time. Can I only apply it to displacement-time data? Thank you.

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Yes, at least if the ringing is sinusoidal.

The derivatives of a sinusoid, i.e. velocity and acceleration, are still sinusoids. The magnitude of the derivatives will be enlarged in proportion to the frequency, frequency squared, etc.

As long as the frequency stays the same, the amplitude-envelope of the ringing will decay with the same exponential time constant.

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