Including Hysteresis
Since the compressed hammer-felt (wool) on real piano hammers shows significant hysteresis memory, an improved piano-hammer felt model is
where



Equation (9.21) is said to be a good approximation under normal playing conditions. A more complete hysteresis model is [487]
![$\displaystyle f_h(t) \eqsp f_0\left[x_k^p(t) - \frac{\epsilon}{\tau_0} \int_0^t x_k^p(\xi) \exp\left(\frac{\xi-t}{\tau_0}\right)d\xi\right]
$](http://www.dsprelated.com/josimages_new/pasp/img2179.png)

Relating to Eq.(9.21) above, we have
(N/mm
).
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Nonlinear Spring Model