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Homework Help: Estimating the damping ratio from the waveform graph

  1. Jul 27, 2013 #1
    1. The problem statement, all variables and given/known data

    From the waveform shown below, estimate
    a) the damping ratio ζ (you may compare response with a standard chart);
    b) the forced or damped frequency of oscillation; and
    c) the natural or undamped frequency of oscillation.
    2. Relevant equations

    Since the waveform is under damped, I'm attempting to use the logarithmic decrement method, described here: http://en.wikipedia.org/wiki/Logarithmic_decrement

    [tex]\sigma = \frac{1}{n}\ln\frac{x(t)}{x(t + nT)}[/tex]
    [tex]\zeta = \frac{1}{\sqrt{1 + \left(\frac{2\pi}{\sigma}\right)^2}}[/tex]
    [tex]f_d = \frac{1}{T}[/tex]
    [tex]f_n = \frac{f_d}{\sqrt{1 - \zeta^2}}[/tex]
    3. The attempt at a solution

    I have estimated the first two peaks from the graph as:
    [tex]p_1 = 0.438\text{ V} \text{ at } 0.27\text{ ms}[/tex]
    [tex]p_2 = 0.350\text{ V} \text{ at } 0.77\text{ ms}[/tex]

    Using the above equations:
    \sigma &= \ln\left(\frac{p_1}{p_2}\right)\\
    &= \ln\left(\frac{0.438}{0.350}\right)\\
    &= 0.224\\
    \zeta &= \frac{1}{\sqrt{1 + \left(\frac{2\pi}{0.224}\right)^2}}\\
    &= 0.0356\\
    f_d &= \frac{1}{0.77 \times 10^{-3} - 0.27 \times 10^{-3}}\\
    &= 2\text{ kHz}\\
    f_n &= \frac{2000}{\sqrt{1 - 0.0356^2}}\\
    &= 2001\text{ Hz}

    The problem is that I'm not sure I believe the results. I'm trying to verify the results by putting them back into the second order characteristic equation:
    \text{C.E.} &= s^2 + 2\zeta{\omega}_{n}s + {\omega}_{n}^2\\
    &= s^2 + (2 \times 0.0356 \times 2\pi \times f_n)s + (2\pi \times f_n)^2\\
    &= s^2 + 895s + 158071624
    then simulating that with a Laplace block in PSpice. However, the simulated waveform doesn't match the one above. The frequency is correct, but the damping ratio is too low -- playing about with the numbers, I find I need to increase the damping ratio to approximately ##2.8\zeta## to get the waveform looking correct.

    I don't know if there is a problem in my method and the results are wrong, or if my simulation is in error (or possibly both!). Can someone please help?

    Last edited by a moderator: Mar 13, 2017
  2. jcsd
  3. Jul 27, 2013 #2

    rude man

    User Avatar
    Homework Helper
    Gold Member

    I got different results. My fn was about 2009 Hz and my ζ = 0.0974. I estimated fd = 2000 Hz and peak ratio = 1.85.

    I can't check your math since you did not define n and σ. You were aware that x = 0 corresponds to 250 mV, right?

    I did notice that (my ζ)/(your ζ) was about the number you thought it should be.
  4. Jul 28, 2013 #3
    I didn't consider ##x(0)##. I guess it makes sense as the wave seems to be settling to 250 mV, but I don't see how it is relevant. As I understand the method, you estimate based on two successive positive peaks, which I have done.

    Which peaks did you use and what did you estimate their coordinates to be?

    In my calculations I chose the first two consecutive peaks, thus ##n = 1## (I should have been more explicit about that). Why do you say that I haven't defined ##\sigma## though? I did show my working, repeated below:
    \sigma &= \ln\left(\frac{p_1}{p_2}\right)\\
    &= \ln\left(\frac{0.438}{0.35}\right)\\
    &= 0.224
  5. Jul 28, 2013 #4
    Ah! Don't worry, I've figured out where I've gone wrong, helped by your comment about ##x(0)##. I've incorrectly used the absolute peak values, rather than their relative values from ##x(0)##.

    Thanks for the help!
  6. Mar 13, 2017 #5
    Uploading waveform image again, since the link in the original post is now broken and I can't figure out how to edit the post.
  7. Apr 17, 2018 at 5:09 PM #6
    does anyone know how to estimate the x and y axis sensitivities if you were given this plot?
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