Estimating the damping ratio from the waveform graph

[cjs94]

Homework Statement

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.

Homework 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

$$\sigma = \frac{1}{n}\ln\frac{x(t)}{x(t + nT)}$$
$$\zeta = \frac{1}{\sqrt{1 + \left(\frac{2\pi}{\sigma}\right)^2}}$$
$$f_d = \frac{1}{T}$$
$$f_n = \frac{f_d}{\sqrt{1 - \zeta^2}}$$

The Attempt at a Solution

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

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

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:
$$\begin{align}<br /> \text{C.E.} &= s^2 + 2\zeta{\omega}_{n}s + {\omega}_{n}^2\\<br /> &= s^2 + (2 \times 0.0356 \times 2\pi \times f_n)s + (2\pi \times f_n)^2\\<br /> &= s^2 + 895s + 158071624<br /> \end{align}$$
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?

Thanks,
Chris

 

[rude man]

I got different results. My f_n was about 2009 Hz and my ζ = 0.0974. I estimated f_d = 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.

 

[cjs94]

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:
$$\begin{align}<br /> \sigma &= \ln\left(\frac{p_1}{p_2}\right)\\<br /> &= \ln\left(\frac{0.438}{0.35}\right)\\<br /> &= 0.224<br /> \end{align}$$

 

[cjs94]

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!

 

[cjs94]

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.

 

[Colton Knight]

does anyone know how to estimate the x and y-axis sensitivities if you were given this plot?