Nonlinear Spring Model

In the musical acoustics literature, the piano hammer is classically modeled as a nonlinear spring [493,63,178,76,60,486,164].^10.14Specifically, the piano-hammer damping in Fig.9.22 is typically approximated by , and the spring is nonlinear and memoryless according to a simple power law:

where for a linear spring, and generally for pianos. A fairly complete model across the piano keyboard (based on acoustic piano measurements) is as follows [487]:

The upward force applied to the string by the hammer is therefore

This force is balanced at all times by the downward string force (string tension times slope difference), exactly as analyzed in §9.3.1 above.

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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.