### 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).

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].

^{10.14}Specifically, 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:

(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]

#### Piano Hammer Mass

The piano-hammer*mass*may be approximated across the keyboard by [487]

**Next Section:**

Pluck Modeling

**Previous Section:**

Ideal String Struck by a Mass