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Nonlinear Spring Model

In the musical acoustics literature, the piano hammer is classically modeled as a nonlinear spring [496,63,179,76,60,489,165].10.14Specifically, the piano-hammer damping in Fig.9.22 is typically approximated by $ \mu=0$ , and the spring $ k$ is nonlinear and memoryless according to a simple power law:

$\displaystyle k(x_k) \; \approx \; Q_0\, x_k^{p-1}

where $ p=1$ for a linear spring, and generally $ p>2$ for pianos. A fairly complete model across the piano keyboard (based on acoustic piano measurements) is as follows [490]:

Q_0 &=& 183\,e^{0.045\,n}\\
p &=& 3.7 + 0.015\,n\\
n &=& \mbox{piano key number $n\in[1,88]$}\\
x_k &=& \mbox{hammer-felt (nonlinear spring) compression}\\
v_k &=& \dot{x}_k

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

$\displaystyle f_h(t) \eqsp Q_0\, x_k^p.$ (10.20)

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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University