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Including Hysteresis

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

$\displaystyle f_h(t) \eqsp Q_0\left[x_k^p + \alpha \frac{d(x_k^p)}{dt}\right], \protect$ (10.21)

where $ \alpha = 248 + 1.83\,n - 5.5 \cdot 10^{-2}n^2$ ($ \mu $ s), and again $ n$ denotes piano key number [489].

Equation (9.21) is said to be a good approximation under normal playing conditions. A more complete hysteresis model is [489]

$\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]
$

where

\begin{eqnarray*}
f_0 &=& \mbox{ instantaneous hammer stiffness}\\
\epsilon &=& \mbox{ hysteresis parameter}\\
\tau_0 &=& \mbox{ hysteresis time constant}.
\end{eqnarray*}

Relating to Eq.$ \,$ (9.21) above, we have $ Q_0=f_0\cdot(1-\epsilon)$ (N/mm$ \null^p$ ).


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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