In [267,268], an extension of the mass-spring model of  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 . The vector equation of motion for mass 1 is given by Newton's second law :
and similarly for mass 2, where is the vector position of mass 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 .
The force on the
th mass can be expressed as