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Physical Audio Signal Processing
    Wave Digital Filters
       Wave Digital Modeling Examples
          Wave Digital Mass-Spring Oscillator
             Oscillation Frequency


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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$.

Previous: Wave Digital Mass-Spring Oscillator
Next: DC Analysis of the WD Mass-Spring Oscillator

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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