Damped Driven Oscillator

1. Sep 8, 2009

swuster

1. The problem statement, all variables and given/known data
A driven oscillator with mass m, spring constant k, and damping coefficient b is is driven by a
force $$F_{o}cos(\omega t)$$. The resulting steady-state oscillations are described by $$x(t) = Re{\underline{A}e^{i\omega t}}$$ where:

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

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

2. Relevant equations
n/a

3. The attempt at a solution
The amplitude is $$Ae^{i\varphi}$$ which has its maximum magnitude when $$\varphi = -\pi/2$$. 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 $$\frac{F_{0}/m}{\omega_{o}^{2}}$$.....

Last edited: Sep 8, 2009