Wave Digital Mass-Spring Oscillator

Let's look again at the mass-spring oscillator of §F.3.4, but this time without the driving force (which effectively decouples the mass and spring into separate first-order systems). The physical diagram and equivalent circuit are shown in Fig.F.32 and Fig.F.33, respectively.

Note that the mass and spring can be regarded as being in parallel or in series. Under the parallel interpretation, we have the WDF shown in Fig.F.34 and Fig.F.35.^F.5 The reflection coefficient can be computed, as usual, from the first alpha parameter:

This result, , is just the ``impedance step over impedance sum'', so no calculation was really necessary.

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Subsections

• Oscillation Frequency
• DC Analysis of the WD Mass-Spring Oscillator
• WD Mass-Spring Oscillator at Half the Sampling Rate
• Linearly Growing State Variables in WD Mass-Spring Oscillator
• A Signal Processing Perspective on Repeated Mass-Spring Poles
• Physical Perspective on Repeated Poles in the Mass-Spring System
• Mass-Spring Boundedness in Reality
• Energy-Preserving Parameter Changes (Mass-Spring Oscillator)
• Exercises in Wave Digital Modeling

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Next: Oscillation Frequency

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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.