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

$$R_h(s) \eqsp ms \left\Vert \left(\mu+\frac{k}{s}\right)\right. \eqsp \frac{\mu s^2 + ks}{s^2+\frac{\mu}{m}s+\frac{k}{m}}. \protect$$ (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 [364,351,560,558].