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 $ \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 [487]:

\begin{eqnarray*}
Q_0 &=& 183\,e^{0.045\,n}\\
p &=& 3.7 + 0.015\,n\\
n &=& \mb...
...hammer-felt (nonlinear spring) compression}\\
v_k &=& \dot{x}_k
\end{eqnarray*}

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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Displacement-Wave Simulation