Damped Driven Oscillator

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Homework Statement


A driven oscillator with mass m, spring constant k, and damping coefficient b is is driven by a
force [tex]F_{o}cos(\omega t)[/tex]. The resulting steady-state oscillations are described by [tex]x(t) = Re{\underline{A}e^{i\omega t}}[/tex] where:

[tex]\underline{A} = \frac{F_{0}/m}{(\omega_{o}^{2} - \omega^{2}) + i(2\omega/\tau)} = Ae^{i\varphi} , \omega_{o} = \sqrt{k/m} , \tau \equiv 2m/b [/tex]

Show that for light damping ([tex]Q = \tau \omega_{o} / 2 >> 1[/tex]), the maximum amplitude occurs at approximately [tex]\omega = \omega_{o}[/tex] and that the maximum amplitude is approximately Q times the amplitude for very low driving frequencies.


Homework Equations


n/a


The Attempt at a Solution


The amplitude is [tex]Ae^{i\varphi}[/tex] which has its maximum magnitude when [tex]\varphi = -\pi/2[/tex]. Therefore, the driving frequency must be near the resonant frequency so that the term is completely imaginary and negative, creating a phase angle of -90 degrees. I don't understand what the light damping and quality factor has to do with this, however, nor how to prove the value of the amplitude. The amplitude for low driving frequencies is [tex]\frac{F_{0}/m}{\omega_{o}^{2}}[/tex].....
 
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