steady state output transfer function

Question

Hi If I'm given an input:

$$r(t) = 2 \sin(3t).$$

And If I'm supposed to get the steady state response $$y(t)$$

for the given transfer function

$$G(s) = \frac{1}{(s+1)(s+1)}$$

Is the following correct:

$$|G(jw)| = \frac{1}{\sqrt{w^2+1}\sqrt{w^2+1}}$$

$$y(t) = \frac{2 \sin(3t + \arctan(3)+\arctan(3))}{10}$$

Answer 1

You have $r(t) = {1 \over i } (e^{i 3t} - e^{-i3t})$, hence the steady state output is $y(t) = {1 \over i } (G(3i) e^{i 3t} - G(-3i) e^{-i3t}) = {2 \over 2i } (G(3i) e^{i 3t} - \overline{G(3i) e^{i3t}} ) = 2 \operatorname{im} (G(3i) e^{i 3t})$.

Evaluating gives $y(t) = - {1 \over 25} (3 \cos (3t) + 4 \sin (3t))$.

If we let $\phi = \arctan {3 \over 4}$, we have $y(t) = -2 {1 \over 10} \sin (3t + \phi) = 2 {1 \over 10} \sin (3t + \phi-\pi)$.

Since $\arctan {3 \over 4} + 2\arctan 3 = \pi$, we have $y(t) = 2 {1 \over 10} \sin (3t - 2\arctan 3)$.