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Quality Factor

The energy loss rate of a weakly damped (i.e., $ \nu\ll 2\,\omega_0$ ) harmonic oscillator is conveniently characterized in terms of a parameter, $ Q_f$ , which is known as the quality factor. This quantity is defined to be $ 2\pi$ times the energy stored in the oscillator, divided by the energy lost in a single oscillation period. If the oscillator is weakly damped then the energy lost per period is relatively small, and $ Q_f$ is therefore much larger than unity. Roughly speaking, $ Q_f$ is the number of oscillations that the oscillator typically completes, after being set in motion, before its amplitude decays to a negligible value. For instance, the quality factor for the damped oscillation shown in Figure 7 is $ 12.6$ . Let us find an expression for $ Q_f$ .

As we have seen, the motion of a weakly damped harmonic oscillator is specified by [see Equation (72)]

$\displaystyle x = a {\rm e}^{-\nu t/2} \cos(\omega_1 t-\phi).$ (81)

It follows that

$\displaystyle \dot{x} =- \frac{a\,\nu}{2}\,{\rm e}^{-\nu\,t/2}\,\cos(\omega_1\,t-\phi)- a\,\omega_1\,{\rm e}^{-\nu\,t/2}\,\sin(\omega_1\,t-\phi).$ (82)

Thus, making use of Equation (78), the energy lost during a single oscillation period is

$\displaystyle \Delta E$ $\displaystyle = -\int_{\phi/\omega_1}^{(2\pi+\phi)/\omega_1} \frac{dE}{dt} dt$    
  $\displaystyle = m \nu a^{2}\int_{\phi/\omega_1}^{(2\pi+\phi)/\omega_1}{\rm e}...
...{2} \cos(\omega_1 t-\phi) + \omega_1 \sin(\omega_1 t-\phi)\right]^{ 2} dt.$ (83)

In the weakly damped limit, $ \nu\ll 2\,\omega_0$ , the exponential factor is approximately unity in the interval $ t=\phi/\omega_1$ to $ (2\pi+\phi)/\omega_1$ , so that

$\displaystyle \Delta E \simeq \frac{m \nu a^{2}}{\omega_1}\int_0^{2\pi} \left...
...,\omega_1 \cos\theta \sin\theta + \omega_1^{ 2} \sin^2\theta\right)d\theta,$ (84)

where $ \theta=\omega_1\,t-\phi$ . Thus,

$\displaystyle \Delta E \simeq \frac{\pi\,m\,\nu\,a^{2}}{\omega_1}\,(\nu^2/4+\omega_1^{\,2}) = \pi\,m\,\omega_0^{\,2}\,a^{2}\left(\frac{\nu}{\omega_1}\right),$ (85)

because, as is readily demonstrated,

$\displaystyle \int_0^{2\pi}\cos^2\theta d\theta = \int_0^{2\pi}\sin^2\theta d\theta$ $\displaystyle =\pi,$ (86)
$\displaystyle \int_0^{2\pi}\cos\theta \sin\theta d\theta$ $\displaystyle =0.$ (87)

The energy stored in the oscillator (at $ t=0$ ) is [cf., Equation (16)]

$\displaystyle E = \frac{1}{2} m \omega_0^{ 2} a^{2}.$ (88)

Hence, we obtain

$\displaystyle Q_f = 2\pi\,\frac{E}{\Delta E} = \frac{\omega_1}{\nu}\simeq \frac{\omega_0}{\nu}.$ (89)


next up previous
Next: LCR Circuits Up: Damped and Driven Harmonic Previous: Damped Harmonic Oscillation
Richard Fitzpatrick 2013-04-08