Piano String FDS, Continued

We have

$$\begin{eqnarray*} {\bf y}_{n} ={\bf A_{1}y_{n-1}}+{\bf A_{2}y_{n-2}}+{\bf A_{3}y_{n-3}}+{\bf g}f_n({\bf y}_{n-1}[i_h],y^h_{n-1}) \end{eqnarray*}$$

The string displacement is updated in time by three matrix-vector products, plus a forcing term ${\bf g} f_n({\bf y}_{n-1}[i_h],y^h(n-1))$ :

• The length $N\times 1$ matrix ${\bf g}$ represents the shape of the hammer excitation over the length of the string. The nonzero elements correspond to the width of the hammer relative to that of the string. For greatest accuracy, it should be time varying, but it may be approximated by a constant shape.

• The element ${\bf y}_n[i_h]$ is the string displacement sample closest to the hammer position.

• The scalar function $y^h_n$ denotes hammer position at time $n$ .

• The scalar function $f_n({\bf y}_{n}[i_h],y^h_n)$ sets the amplitude of the hammer force distribution across position at time $n$ .

• The force exerted on the string by the hammer is a nonlinear function of the hammer-string separation $y^h_n - {\bf y}_{n}[i_h]$ .

• The time evolution of the hammer must be computed numerically by a separate finite difference scheme. Typical models include a classical point mass, a nonlinear spring (which gets stiffer when compressed), and a very small amount of damping.

Note that the FDS involves 3 steps of ``lookback.'' It turns out this model goes unstable in the limit as the sampling rate goes to infinity!

• There is no problem at any normal audio sampling rate--the sampling rate must be on the order of hundreds of megahertz to trigger the instability.
• The model can be stabilized at all sampling rates by using instead a two-time-step scheme.
• Watch for forthcoming publications by Stefan Bilbao at CCRMA.
• We'll talk about stability of finite difference schemes later.