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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,

$\displaystyle \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$ .


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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