### 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].^{10.14}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:

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]

**Next Section:**

Pluck Modeling

**Previous Section:**

Ideal String Struck by a Mass