Estimating the damping ratio from the waveform graph

In summary, the damping ratio ζ was estimated to be 0.224. The forced or damped frequency of oscillation was estimated to be 2 kHz. The natural or undamped frequency of oscillation was estimated to be 2001 Hz.
  • #1
cjs94
16
0

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.
img_0041-jpg.114478.jpg

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

[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]

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:
[tex]\begin{align}
\sigma &= \ln\left(\frac{p_1}{p_2}\right)\\
&= \ln\left(\frac{0.438}{0.350}\right)\\
&= 0.224\\
\text{and}\\
\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}
\end{align}
[/tex]

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:
[tex]\begin{align}
\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
\end{align}
[/tex]
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
 
Last edited by a moderator:
Physics news on Phys.org
  • #2
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.
 
  • #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:
[tex]\begin{align}
\sigma &= \ln\left(\frac{p_1}{p_2}\right)\\
&= \ln\left(\frac{0.438}{0.35}\right)\\
&= 0.224
\end{align}
[/tex]
 
  • #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!
 
  • #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.
IMG_0041.jpg
 
  • #6
does anyone know how to estimate the x and y-axis sensitivities if you were given this plot?
 

FAQ: Estimating the damping ratio from the waveform graph

1. What is the damping ratio in relation to a waveform graph?

The damping ratio is a measure of the rate at which an oscillating system, represented by a waveform graph, decreases in amplitude over time. It is a key parameter in understanding the stability and behavior of dynamic systems.

2. How is the damping ratio calculated from a waveform graph?

The damping ratio can be calculated by measuring the logarithmic decrement between two successive peaks of the waveform. The formula for calculating the damping ratio is: ζ = ln(An/An+1)/(2πn), where An and An+1 are the amplitudes of the nth and (n+1)th peaks, and n is the number of cycles between the peaks.

3. What factors can affect the estimation of the damping ratio from a waveform graph?

There are several factors that can affect the accuracy of the damping ratio estimation, including noise in the signal, non-stationarity of the signal, and the choice of peaks used for calculation. It is important to ensure a clean and stable signal and to use appropriate methods for selecting the peaks in order to obtain an accurate estimation of the damping ratio.

4. What is a typical range for the damping ratio in real-world systems?

The damping ratio can vary depending on the type of system and its characteristics. In general, a damping ratio of less than 1 indicates an underdamped system, while a damping ratio of greater than 1 indicates an overdamped system. Real-world systems typically have damping ratios between 0.01 and 1, with values closer to 1 indicating a higher level of damping.

5. How is the damping ratio used in practical applications?

The damping ratio is a crucial parameter in many engineering and scientific applications. It is used to characterize the behavior of various systems, such as mechanical, electrical, and acoustic systems, and to design control systems for these systems. It is also used in the analysis and design of structures, such as buildings and bridges, to ensure their stability and safety.

Similar threads

Back
Top