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:
![$\displaystyle k(x_k) \; \approx \; Q_0\, x_k^{p-1}
$](http://www.dsprelated.com/josimages_new/pasp/img2172.png)
![$ p=1$](http://www.dsprelated.com/josimages_new/pasp/img2173.png)
![$ p>2$](http://www.dsprelated.com/josimages_new/pasp/img2174.png)
![\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*}](http://www.dsprelated.com/josimages_new/pasp/img2175.png)
The upward force applied to the string by the hammer is therefore
![]() |
(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