High-Accuracy Piano-String Modeling

In [267,268], an extension of the mass-spring model of [394] 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.

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

$$\vert\vert\,\underline{f}_1\,\vert\vert \eqsp k\cdot\vert l_1-l_0\vert \eqsp \vert\vert\,\underline{f}_2\,\vert\vert$$

where $l_0$ denotes the rest-length of the spring $k$ , and $\vert\vert\,\underline{f}_i\,\vert\vert$ denotes the vector norm (length) of the 3D vector $\underline{f}_i\in\mathbb{R}^3$ [454]. The vector equation of motion for mass 1 is given by Newton's second law $f=ma$ :

$$\begin{eqnarray*} m_1\, \underline{{\ddot x}}_1 \eqsp \underline{f}_1 &=& k\cdot\left(\left\Vert\,\underline{x}_2-\underline{x}_1\,\right\Vert-l_0\right)\cdot\frac{\underline{x}_2-\underline{x}_1}{\left\Vert\,\underline{x}_2-\underline{x}_1\,\right\Vert}\\ [5pt] &=& k\left[1-\frac{l_0}{\left\Vert\,\underline{x}_2-\underline{x}_1\,\right\Vert}\right]\left(\underline{x}_2-\underline{x}_1\right) \end{eqnarray*}$$

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

Generalizing to a chain of masses and springs is shown in Fig.9.26. Mass-spring chains--also called beaded strings--have been analyzed in numerous textbooks (e.g., [297,321]), and numerical software simulation is described in [394].

Figure 9.26: Mass-spring string model

The force on the $i$ th mass can be expressed as

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

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

where

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