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:
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 (s), and again 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]
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]
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Pluck Modeling
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Ideal String Struck by a Mass