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1. The problem statement, all variables and given/known data
Transfer function:
G(s) =[tex]\frac{k}{s^{2}+2\varsigma\omega_{n}s+\omega_{n}^{2}}[/tex]
a) if u(t) = cos(2t), find the steady state response
b) determine the values for k, damping ratio ([tex]\varsigma[/tex]), and the natural frequency.
2. Relevant equations
3. 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/[tex]\sqrt{2}[/tex], 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) =[tex]\frac{10}{s^{2}+2.1s+100}[/tex]
when I find the bode plot of this function it looks similar to the provided plot, but the magnitude is off.
Transfer function:
G(s) =[tex]\frac{k}{s^{2}+2\varsigma\omega_{n}s+\omega_{n}^{2}}[/tex]
a) if u(t) = cos(2t), find the steady state response
b) determine the values for k, damping ratio ([tex]\varsigma[/tex]), and the natural frequency.
2. Relevant equations
3. 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/[tex]\sqrt{2}[/tex], 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) =[tex]\frac{10}{s^{2}+2.1s+100}[/tex]
when I find the bode plot of this function it looks similar to the provided plot, but the magnitude is off.
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