damping factor and dampted frequency

Discussion in 'Homework Help' started by ham3388, Oct 3, 2012.

Jul 3, 2012
97
3
Hi friends...

The below is a question about damped frequency,I have done it only I want to check that its correct or not.

Any help from you will be appriciated .

My regards

2. Audioguru New Member

Dec 20, 2007
9,411
896

Why bother to calculate the damping factor since it is VERY low, maybe 2.
Most audio amplifiers today have a better damping factor of 200 to thousands so the output does not resonate.

3. mlog Member

Feb 11, 2012
276
36
I tried to read it, but I really couldn't. Wasn't very large or clear.

4. Audioguru New Member

Dec 20, 2007
9,411
896
I used Microsoft Paint program to add some contrast to the image that was grey on grey.

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5. t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
From the rather fuzzy graph I obtained 4 pairs of (time,value) data

t=0.26 ms V=0.44 V
t=0.76 ms V=0.35 V
t=0.52 ms V=0.12 V
t=1.02 ms V=0.18 V

From these data and some maths [not shown] it's possible to show the steady state final value is ~250mV. This is also fairly obvious from the graph trend to a steady-state final value of that order.

One may then find the decay time factor a=1/τ using

$ln$$\frac{0.44-0.25}{0.35-0.25}$$=a$$0.76-0.26$$\times 10^{-3}$

or

$a=\frac{1}{$$0.76-0.26$$\times 10^{-3}} $ln$$\frac{0.44-0.25}{0.35-0.25}$$$=2000 \times ln{$$\frac{0.19}{0.1}$$}$

But
$a=\zeta \omega_o$

where
$\omega_o=\sqr{{\omega_d}^2+a^2}$

Using the above data the damped period is clearly Td= 0.5ms

Hence

$\omega_d=\frac{2\pi}{T_d}=\frac{2\pi}{0.5\times10^{-3}}=12566 \ rad \ sec^{-1}$

Which is the same as the OP's result.

This should provide sufficient alternative data to enable a cross check of the OP's working - which BTW looks to have some errors. For instance - a damping factor ζ of 0.86 would probably not produce the extended oscillatory result shown in the graph.

Last edited: Oct 4, 2012

Jul 3, 2012
97
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Thanks a looooooot t n k .

Im really confused, I dont know where is my mistake since I have done it according to the formulae.

Can you guide me please .

Jul 3, 2012
97
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I think the Vpeak is wrongly calculated.

Vpeak =430-225=205 mV as per my calculation.

If its so, then whats it suppose to be ?

8. t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
Perhaps it would be worth establishing a common basis for analysis - otherwise we might be at cross purposes.

I'm basing an analysis on the typical second order characteristic polynomial of the form

$s^2+2as+{\omega_o^2}$

where damping factor ζ fits according to the relationship

$a=\zeta \omega_o$

and the damped frequency is related to a and ωo by

$\omega_d^2=\omega_o^2-a^2$

Is this what your equations would be based upon?

Jul 3, 2012
97
3
I will post the standart formulae of Damping factor by today evening since Im away from the books right now.
My regards

10. t_n_k AAC Fanatic!

Mar 6, 2009
5,448
783
It's such a poor image it's hard to extract the actual values. For my eye, the first peak is around 440mV and the steady state final value is about 250mV. So the difference would give Vpeak=190mV. But that probably wouldn't account for the disparity in our respective damping factor calculations.

Jul 3, 2012
97
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Then I dont remember any other method for calculating Vpeak.

In other words , how to calculate the Vpeak value?

Jul 3, 2012
97
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Then how I have to proceed instead?

Jul 3, 2012
97
3
So I will come back to you once I refer to the books

cheers

14. t_n_k AAC Fanatic!

Mar 6, 2009
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The question doesn't really spell out the nature of the damped oscillatory response.

One might consider it to be typical of the voltage step [input step magnitude E volts] response of a series RLC circuit with the output across the capacitor.

Such a response is of the form

$V_c(t)=E $1-\frac{\omega_o}{\omega_d}e^{-at}\sin{$$\omega_dt+\phi$$}$$

The damping term a may be found from

$a=-\frac{2}{T_d}\ln{$$\frac{V_{peak}-V_{\infty}}{V_{\infty}}$$}$

Where Td is the damped frequency period and V∞ is the output steady-state value - these values [including Vpeak] must be determined from direct measurements performed on the output response plot.

From this one may deduce the natural frequency ωo as ..

$\omega_o=\sqr{\omega_d^2 + a^2}$

Where

$\omega_d=\frac{2\pi}{T_d}$

Finally the damping factor ζ can be found from ...

$\zeta=\frac{a}{\omega_o}$

Jul 3, 2012
97
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Hi again....

I have uploaded attachemnt that shows the method I used

please check and let me know

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16. mlog Member

Feb 11, 2012
276
36
I see several things wrong. Some of it is simply "eye calibration."

1. The peak is 430 mV
2. The SS value is 250 mV
3. The delta peak = 430-250 = 180 mV
4. The ratio of delta peak to SS is 180/250 = 0.720
5. Your equation for damping ratio is missing a square.
6. Damping Ratio = { 1+ [ln(∏/0.720)]^2 }^(-0.5) = 0.104
7. Peak time appears to be 0.26 ms.
8. Damped frequency = ∏ / 0.000,26 = 12,080 rad/s
9. Natural frequency = (12,080) * sqrt(1-0.104^2) = 12,030 rad/s

You know the damping ratio can't be close to 0.86, because that is nearly critically damped and would have almost zero overshoot. Snce the damping ratio is really about 0.1, it means the damped and natural frequencies are nearly the same -- which they are within 0.5% of each other.

One final comment. You might notice I'm not using a lot of significant figures. In fact, I might be pushing it a little with 4 digits on the frequencies. Since your measurements are based on the human eye reading a plot, you can only guestimate the values. For exmple, if the peak is judged to be 430, how can I be certain it isn't 429 or even 433? There is an error to what we can read. Does it make sense to say the damping ratio is 0.103,999? It makes more sense to say it's 0.10 or 0.104 at best. I'm mentioning this because of your original frequency calculations, e.g. 12,566.

17. t_n_k AAC Fanatic!

Mar 6, 2009
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It's a pain often trying to pin down these wretched equations ...

The actual equation should read ....

$\LARGE{\zeta=\frac{1}{\sqr{1 + {$$\frac{\pi}{\ln{ \( \frac{V_{pk} - V_{SS}}{V_{SS}}$$ }}\)}^2 }}}$

Where Vpk is the output response peak value and Vss is the steady-state [final] output value

ham3388 likes this.
18. mlog Member

Feb 11, 2012
276
36
Yeah, that's right. Looks better in LaTex or whatever you call it. I had the equation correct before I transferred it from paper to keyboard. I admire your ability to be able to use it so well. I really struggle to write equations with that program.

ham3388 likes this.