Nonlinear Piano-String Equations of Motion in State-Space Form

For the flexible (non-stiff) mass-spring string, referring to Fig.9.26 and Eq.(9.34), we have the following equations of motion:

$$\begin{eqnarray*} \underline{f}_1 &=& m_1 \,\underline{{\ddot x}}_1 \eqsp \alpha_1\cdot(\underline{x}_2-\underline{x}_1)\\ \vdots && \vdots\\ \underline{f}_i &=& m_i \,\underline{{\ddot x}}_i \eqsp \alpha_{i-1}\,\underline{x}_{i-1} - (\alpha_{i-1}+\alpha_i)\,\underline{x}_i + \alpha_i\,\underline{x}_{i+1}\\ \vdots && \vdots\\ \underline{f}_M &=& m_M \,\underline{{\ddot x}}_M \eqsp \alpha_{M-1}\cdot(\underline{x}_{M-1} - \underline{x}_M) \end{eqnarray*}$$

or, in $3M\times1$ vector form,

$$\underline{F}\eqsp \mathbf{M}\, \ddot{\underline{X}} \eqsp \mathbf{A}\, \underline{X}.$$

Here the string terminations (bridge and agraffe) are modeled simply as very large masses $m_1$ and $m_M$ .