Nonlinear Spring Model

In the musical acoustics literature, the piano hammer is classically modeled as a nonlinear spring [496,63,179,76,60,489,165].^10.14Specifically, the piano-hammer damping in Fig.9.22 is typically approximated by $\mu=0$ , and the spring $k$ is nonlinear and memoryless according to a simple power law:

$$k(x_k) \; \approx \; Q_0\, x_k^{p-1}$$

where $p=1$ for a linear spring, and generally $p>2$ for pianos. A fairly complete model across the piano keyboard (based on acoustic piano measurements) is as follows [490]:

$$\begin{eqnarray*} Q_0 &=& 183\,e^{0.045\,n}\\ p &=& 3.7 + 0.015\,n\\ n &=& \mbox{piano key number $n\in[1,88]$}\\ x_k &=& \mbox{hammer-felt (nonlinear spring) compression}\\ v_k &=& \dot{x}_k \end{eqnarray*}$$

The upward force applied to the string by the hammer is therefore

$$f_h(t) \eqsp Q_0\, x_k^p.$$ (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.