High-Accuracy Piano-String Modeling

In [265,266], an extension of the mass-spring model of [391] was presented for the purpose of high-accuracy modeling of nonlinear piano strings struck by a hammer model such as described in §9.3.2. This section provides a brief overview.

**Figure 9.25:** Mass-spring model in 3D space.
$\includegraphics[width=0.35\twidth]{eps/massspringmass}$

Figure 9.25 shows a mass-spring model in 3D space. From Hooke's Law (§B.1.3), we have

$\displaystyle \vert\vert\,\underline{f}_1\,\vert\vert \eqsp k\cdot\vert l_1-l_0\vert \eqsp \vert\vert\,\underline{f}_1\,\vert\vert$

where

denotes the rest-length of the spring

, and $\vert\vert\,\underline{f}_i\,\vert\vert$ denotes the vector norm (length) of the 3D vector $\underline{f}_i\in{\bf R}^3$ [451]. The vector equation of motion for mass 1 is given by Newton's second law

$\begin{eqnarray*} m_1\, \underline{{\ddot x}}_1 \eqsp \underline{f}_1 &=& k\cdo... ...,\right\Vert}\right]\left(\underline{x}_2-\underline{x}_1\right) \end{eqnarray*}$

and similarly for mass 2, where $\underline{x}_i\in{\bf R}^3$ is the vector position of mass in 3D space.

Generalizing to a chain of masses and spring is shown in Fig.9.26. Mass-spring chains--also called beaded strings--have been analyzed in numerous textbooks (e.g., [295,318]), and numerical software simulation is described in [391].

**Figure 9.26:** Mass-spring string model
$\includegraphics[width=0.8\twidth]{eps/massspringstring}$

The force on the th mass can be expressed as

$\displaystyle \underline{f}_i$	$\displaystyle =$	$\displaystyle \alpha_i\cdot\left(\underline{x}_{i+1}-\underline{x}_i\right) + \alpha_{i-1}\cdot\left(\underline{x}_{i-1}-\underline{x}_i\right)$
	$\displaystyle =$	$\displaystyle \alpha_{i-1}\,\underline{x}_{i-1} - (\alpha_{i-1}+\alpha_i)\,\underline{x}_i + \alpha_i\,\underline{x}_{i+1} \protect$	(10.34)

where

$\displaystyle \alpha_i \isdefs k\cdot \left[1-\frac{l_0}{\left\Vert\,\underline{x}_{i+1}-\underline{x}_i\,\right\Vert}\right].$

Subsections

Previous: Checking the Approximations
Next: A Stiff Mass-Spring String Model

Order a Hardcopy of Physical Audio Signal Processing

About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

Sign in

Search Online Books

Free Online Books

Sponsor

Chapters

See Also

Physical Audio Signal Processing
    Virtual Musical Instruments
       Piano Synthesis
          High-Accuracy Piano-String Modeling