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

where denotes the rest-length of the spring , and denotes the vector norm (length) of the 3D vector [454]. The

and similarly for mass 2, where 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.*, [297,321]), and numerical software
simulation is described in [394].

The force on the
th mass can be expressed as

where

- A Stiff Mass-Spring String Model
- Nonlinear Piano-String Equations of Motion in State-Space Form
- Finite Difference Implementation

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