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].

---

Subsections

• Nonlinear Spring Model
• Including Hysteresis
• Piano Hammer Mass

---

Previous: Displacement-Wave Simulation
Next: Nonlinear Spring Model

Order a Hardcopy of Physical Audio Signal Processing

---

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.