Finding gain from a bode plot

[snoweangel27]

Homework Statement

Transfer function:
G(s) =$$\frac{k}{s^{2}+2\varsigma\omega_{n}s+\omega_{n}^{2}}$$

a) if u(t) = cos(2t), find the steady state response
b) determine the values for k, damping ratio ($$\varsigma$$), and the natural frequency.

Homework Equations

The Attempt at a Solution

I know how to find the steady state response by finding G(wj) where w = 2 from the input function. However, I can't seem to find the correct values for k, damping ratio, and the natural frequency.

first I found the cut off magnitudes from the bode plot by multiplying the largest magnitude (40) by (1/$$\sqrt{2}$$, to obtain the magnitude of the cut off frequencies to be approx. 28.28db. and found the cut off frequencies to be 8 and 10.1 rad/sec, which allows the Bandwidth to be determined by subtracting the two cut off frequencies.

I believe the natural frequency should be 10, since it is also the center frequency, and I found the damping ration to be 0.105 by dividing the bandwidth by 2 times the natural frequency. And I think the gain should be 10.

my transfer function
G(s) =$$\frac{10}{s^{2}+2.1s+100}$$

when I find the bode plot of this function it looks similar to the provided plot, but the magnitude is off.

 

[Päällikkö]

It's been a while I've done these, hopefully I'm not terribly wrong here as it's not quite my area of specialty. At least I got the image plotted in Matlab and it is indeed exactly the same.

Anyway, setting s = 0, you immediately see that the low pass gain ought to be k/w_n^2. Yours is 1/10, which, if my math is right, gives -20 dB gain rather than 20 dB as shown in the figure.

Phase -90 is quite special. Use that to get a second relation between the parameters. Finally I'd proceed to check what value and where |G| gets as its maximum to find the third and last relation between the parameters. You ought to get nice integers as answers (well, the damping parameter is a reciprocal of one).

 

[Mene]

I am not sure where I went wrong.



It looks like you have made a good attempt at finding the gain from the bode plot and determining the values for k, damping ratio, and the natural frequency. However, there may be some errors in your calculations that are causing the discrepancy in your results. It is important to double check your calculations and make sure you are using the correct equations and values.

One potential error could be in your determination of the cut off frequencies. It is possible that the values you have calculated are slightly off, which could affect your results for the natural frequency and damping ratio. Additionally, it is important to make sure you are using the correct units for the cut off frequencies, as this could also affect your calculations.

If you are still having trouble finding the correct values, it may be helpful to consult with a classmate or the instructor for further clarification. It is also a good idea to check your work against any provided solutions or ask for feedback from the instructor to ensure that you have correctly completed the assignment. With some additional effort and attention to detail, you should be able to find the correct values and gain from the bode plot. Keep up the good work!