Piano Hammer Modeling
The previous section treated an ideal point-mass striking an ideal string. This can be considered a simplified piano-hammer model. The model can be improved by adding a damped spring to the point-mass, as shown in Fig.9.22 (cf. Fig.9.12).
![]() |
The impedance of this plucking system, as seen by the string, is the
parallel combination of the mass impedance and the damped spring
impedance
. (The damper
and spring
are formally
in series--see §7.2, for a refresher on series versus
parallel connection.) Denoting
the driving-point impedance of the hammer at the string contact-point
by
, we have
Thus, the scattering filters in the digital waveguide model are second order (biquads), while for the string struck by a mass (§9.3.1) we had first-order scattering filters. This is expected because we added another energy-storage element (a spring).
The impedance formulation of Eq.(9.19) assumes all elements are
linear and time-invariant (LTI), but in practice one can normally
modulate element values as a function of time and/or state-variables
and obtain realistic results for low-order elements. For this we must
maintain filter-coefficient formulas that are explicit functions of
physical state and/or time. For best results, state variables should
be chosen so that any nonlinearities remain memoryless in the
digitization
[361,348,554,555].
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$](http://www.dsprelated.com/josimages_new/pasp/img2174.png)
![\begin{eqnarray*}
Q_0 &=& 183\,e^{0.045\,n}\\
p &=& 3.7 + 0.015\,n\\
n &=& \mb...
...hammer-felt (nonlinear spring) compression}\\
v_k &=& \dot{x}_k
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2175.png)
The upward force applied to the string by the hammer is therefore
![]() |
(10.20) |
This force is balanced at all times by the downward string force (string tension times slope difference), exactly as analyzed in §9.3.1 above.
Including Hysteresis
Since the compressed hammer-felt (wool) on real piano hammers shows significant hysteresis memory, an improved piano-hammer felt model is
where
![$ \alpha = 248 + 1.83\,n - 5.5 \cdot 10^{-2}n^2$](http://www.dsprelated.com/josimages_new/pasp/img2178.png)
![$ \mu $](http://www.dsprelated.com/josimages_new/pasp/img60.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
Equation (9.21) is said to be a good approximation under normal playing conditions. A more complete hysteresis model is [487]
![$\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]
$](http://www.dsprelated.com/josimages_new/pasp/img2179.png)
![\begin{eqnarray*}
f_0 &=& \mbox{ instantaneous hammer stiffness}\\
\epsilon &=&...
...resis parameter}\\
\tau_0 &=& \mbox{ hysteresis time constant}.
\end{eqnarray*}](http://www.dsprelated.com/josimages_new/pasp/img2180.png)
Relating to Eq.(9.21) above, we have
(N/mm
).
Piano Hammer Mass
The piano-hammer mass may be approximated across the keyboard by [487]
![$\displaystyle m = 11.074 - 0.074\,n + 0.0001\,n^2.
$](http://www.dsprelated.com/josimages_new/pasp/img2183.png)
![$ n$](http://www.dsprelated.com/josimages_new/pasp/img146.png)
Next Section:
Pluck Modeling
Previous Section:
Ideal String Struck by a Mass