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I'm an engineering coop at a vibrations consultant. I was given a graph plotting a series of applied frequencies against a floor's acceleration response to them. I was asked to find the natural frequency and damping ratio, but I haven't been able to find the damping ratio. I've taken a course in vibrations, but we were always given more about the scenario (mass of the object, spring constant, etc.). I've asked him to at least give me a starting point, but he's refused.

My boss also asked me to find the theory behind how the damping ratio is determined empirically. All the resources I've found online either reiterate what I already learned in my courses, or are so advanced that I can't even begin to understand them.

My questions are: how is the damping ratio determined empirically, and what is the general way to solve this problem?

For context, my vibrations course focused on 1 DOF systems, teaching the basics behind damping ratio, natural frequency, and general vibrations. We did questions about base excitation and rotating imbalances in machinery.

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    $\begingroup$ What does: "I was given a graph plotting a series of applied frequencies against a floor's acceleration response to them" mean exactly? Do you have a frequency response function (both amplitude and phase or amplitude only) describing the systems response? Do you have the raw data behind the graph? Anyway, for a quick estimate (I can write an answer later) google: Q-factor and/or 3dB bandwidth. $\endgroup$
    – user883521
    Jul 5, 2017 at 5:26

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The Wikipedia page is somewhat electrically focused, but it should be good enough to get you going.

Step 1: determine the frequency at which the response peaks, call it fc

step 2: determine the frequencies at which the power is 1/2 of the peak power (i.e. amplitude is sqrt(2)/2 times the peak). there are 2 such frequencies, one higher than the peak and one lower. call them f2 and f1

step 3: Q=fc / (f2-f1), and damping ratio zeta = 1/(2*Q)

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  • $\begingroup$ freq vs.amplitude. This can find harmonic fre $\endgroup$
    – RainerJ
    Jul 5, 2017 at 11:27
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    $\begingroup$ This gives a reasonable estimate for light damping of well separated modes, but in general the measured response curve is affected by all the modes, not just the "nearest" one. Google for "modal analysis system identification" for more. If you only have a single response curve, the next level up from half-power bandwidth is a circle-fitting method on a Nyquist plot, if you measured both the amplitude and phase of the response. $\endgroup$
    – alephzero
    Jul 5, 2017 at 12:14

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