PX153 - E3 - driven damped simple harmonic motion

PX155 - D9 - driven damped simple harmonic motion

• add a periodic driving force to our damped SHM. we'll write

- often written as $A_0\cos (\omega t)$

• the equation of motion becomes:

- we'll assume underdamped, so the complementary function is
$$x_{c}= e^{-\alpha t} (Ce^{i\omega't}+De^{-i\omega't})$$
- where, ${\omega }^{\prime }=\sqrt{{\omega }_0^2-{\alpha }^2}$
- for the particular integral, our trial function is $x_p(t)=a{e}^{i\omega t}$
- $a$ could be a complex number
$-{\omega }^{2}a{e}^{i\omega t}+2\alpha i\omega a{e}^{i\omega t}+{\omega }_0^{2}a{e}^{i\omega t}=A_0{e}^{i\omega t}$
$a({\omega }_0^2-{\omega }^2+2\alpha i\omega )=A_0$
$\therefore a=\frac{A_0}{({\omega }_0^2-{\omega }^2)+2\alpha i\omega }$
- $a$ is a complex number
$|a|=\sqrt{a^{\ast }a}=\sqrt{\frac{A_0}{({\omega }_0^2-{\omega }^2)+2\alpha i\omega }}=A$
$arg(a)={\tan }^{-1}\frac{Im(a)}{Re(a)}$
$arg(a)=\varphi ={\tan }^{-1}(\frac{-2\alpha \omega }{{\omega }_0^2-{\omega }^2})$
- find maximum of $a$ and sketch
$$G.S. : x(t) = e^{-\alpha t} (Ce^{i\omega't} + De^{-i\omega't}) + Ae^{i(\omega t + \phi)}$$
- where, $A=\sqrt{\frac{A_0}{({\omega }_0^2-{\omega }^2)+2\alpha i\omega }}$ and $\varphi ={\tan }^{-1}(\frac{-2\alpha \omega }{{\omega }_0^2-{\omega }^2})$