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Embedded SystemsFPGAElectronics

Physical Audio Signal Processing
    Virtual Musical Instruments
       String Excitation
          Piano Hammer Modeling
             Including Hysteresis

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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 [487].

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] $

where

\begin{eqnarray*} f_0 &=& \mbox{ instantaneous hammer stiffness}\\ \epsilon &=&... ...resis 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$).

Previous: Nonlinear Spring Model
Next: Piano Hammer Mass

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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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