Problem 1. Resonance (30 points) Consider the following...

Problem 1. Resonance (30 points) Consider the following system: y(t) + y(t) = u(t). Let u(t) = sin(?t), t ? 0, where ? > 0. Can resonance occur in this system? If yes, obtain the resonance frequency. If not, explain. Problem 2. Sinusoidal Transfer Function (50 points) Obtain frequency response for the following systems when u(t) = sin 2t, t ? 0 • G(s) = \frac{1}{s+1} • G(s) = \frac{1}{s^2+3s+2} Problem 3. Sinusoidal Transfer Function (20 points) The following figure shows frequency response of the stable system G(s), where the dashed line is the input u(t) and the solid line is the output y(t). Compute |G(i?)| and ?G(i?).

Transcript

00:01 Hello students, in this question given as a transfer function that is h of s is equals to t upon es square plus fs plus g.

00:23 The first condition is given as when s is equals to 0 then h of s value is equals to the 4.

00:30 If we substitute the value of s is equals to 0 in above equation above transfer function so t upon g will be equal to the 4.

00:44 Now this is equation number 1.

00:47 Let we have to write the transfer function in another form.

00:54 We will divide by the e so this will be equal to the 1 upon s square plus f upon e into s plus g upon e.

01:07 Now we will compare this denominator value by the standard equation that is equals to the s square plus 2 chi omega n s plus omega n square.

01:26 Now after comparing the denominator value we can write f upon e is equals to 2 chi omega n and g upon e is equals to omega n square.

01:46 Now next boundary condition is given as settling time for 2 % criterion is equals to the given as the settling time is equals to 4 by 3 which is equals to the 4 upon chi omega n.

02:03 So chi omega n will be equal to the 3 and the third condition is given as the damping ratio chi is given as 0 .5 radian per second.

02:18 So omega n will be equal to the 6 radian per second.

02:26 Now we can write f upon e is equals to omega n square f upon e is equals to chi omega n 2 times of this so this will be equal to the 6 equation number 2 and g upon e is equals to omega n square that will be equal to the 36 equation number.

02:56 Now we can write d upon e is equals to d upon e is equals to d upon g into g upon e...

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