Old spec 5.3 worked example, first order response to a harmonic input by Laplace (ENGR5001)

Old spec 5.3 worked example, first order response to a harmonic input by LaplaceTutorialOld spec ENGR2022:115 min

ENGR202 (old spec) Control 5.3 Frequency Response Revisited, first order response to a harmonic input (slides 5-7)

The first order system $\tau\dfrac{dx}{dt} + x = Ku(t)$ , initially at rest ( $x(0) = 0$ ), is driven by the harmonic input $u(t) = a\sin(\omega t)$ .

Formulas you may need

$\mathcal{L}[\sin\omega t] = \dfrac{\omega}{s^2 + \omega^2}$ (learn this)
First order frequency response $M = \dfrac{K}{\sqrt{1 + \tau^2\omega^2}}$ , $\phi = -\tan^{-1}(\omega\tau)$ (learn this)

(a)
Find $X(s)$ .
(b)
Show that $x(t)$ is the sum of a decaying transient and a steady state sinusoid, and find the coefficient of the transient term (use the partial fraction for the pole at $s = -1/\tau$ and the condition $x(0) = 0$ as a check).
(c)
For $K = 2$ , $\tau = 4$ s, $a = 1$ and $\omega = 0.5$ rad/s, find the steady state amplitude and phase of the output and the transient coefficient.
steady state amplitude aM =
phase =deg
transient coefficient =