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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).

Figure 9.22: Ideal string excited by a mass and damped spring (a more realistic piano-hammer model).

The impedance of this plucking system, as seen by the string, is the parallel combination of the mass impedance $ ms$ and the damped spring impedance $ \mu+k/s$. (The damper $ \mu $ and spring $ k/s$ 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 $ R_h(s)$, we have

$\displaystyle R_h(s) \eqsp ms \left\Vert \left(\mu+\frac{k}{s}\right)\right. \eqsp \frac{\mu s^2 + ks}{s^2+\frac{\mu}{m}s+\frac{k}{m}}. \protect$ (10.19)

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 $ \mu=0$, and the spring $ k$ is nonlinear and memoryless according to a simple power law:

$\displaystyle k(x_k) \; \approx \; Q_0\, x_k^{p-1}

where $ p=1$ for a linear spring, and generally $ p>2$ for pianos. A fairly complete model across the piano keyboard (based on acoustic piano measurements) is as follows [487]:

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

The upward force applied to the string by the hammer is therefore

$\displaystyle f_h(t) \eqsp Q_0\, x_k^p.$ (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

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


f_0 &=& \mbox{ instantaneous hammer stiffness}\\
\epsilon &=&...
...resis parameter}\\
\tau_0 &=& \mbox{ hysteresis time constant}.

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

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.

where $ n$ is piano key number as before.

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Pluck Modeling
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Ideal String Struck by a Mass