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].
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
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.
where (s), and again denotes piano key number .
Relating to Eq.(9.21) above, we have (N/mm).
Ideal String Struck by a Mass