105 Example of gain and phase

For $a,b>0$, $u_0$ a constant and $y(0)=y^{\prime }(0)=0$, find the gain and phase of

$$a{y}^{\prime \prime }+b{y}^{\prime }+y=u_0\cos \omega t.$$

Then the Laplace transform is

$$(a{s}^2+bs+1)Y(s)=u_0\frac{s}{{\omega }^2+s^2},$$

so that taking $s=i\omega$ in the formula

$$T(s)=\frac{1}{a{s}^2+bs+1},$$

$$T(i\omega )=\frac{1}{1-a{\omega }^2+ib\omega }=\frac{1-a{\omega }^2-ib\omega }{{(1-a{\omega }^2)}^2+b^2{\omega }^2},$$