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Oscillation Frequency

From Fig.F.33, we can see that the impedance of the parallel combination of the mass and spring is given by

$\displaystyle R_{m\vert\vert k}(s) \isdef \left.\frac{k}{s} \right\Vert ms = \frac{\frac{k}{s}ms}{\frac{k}{s}+ms} = \frac{ks}{s^2+\frac{k}{m}} \protect$ (F.38)

(using the product-over-sum rule for combining impedances in parallel). The poles of this impedance are given by the roots of the denominator polynomial in $ s$ :

$\displaystyle s = \pm j\sqrt{\frac{k}{m}} \protect$ (F.39)

The resonance frequency of the mass-spring oscillator is therefore

$\displaystyle \omega_0 = \sqrt{\frac{k}{m}} \protect$ (F.40)

Since the poles $ s=\pm j\omega_0$ are on the $ j\omega $ axis, there is no damping, as we expect.

We can now write reflection coefficient $ \rho$ (see Fig.F.35) as

$\displaystyle \rho = \frac{m-k}{m+k} = \frac{1-\frac{k}{m}}{1+\frac{k}{m}} = \frac{1-\omega_0^2}{1+\omega_0^2}
$

We see that dc ( $ \omega_0=0$ ) corresponds to $ \rho=1$ , and $ \omega_0=\infty$ corresponds to $ \rho=-1$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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