In [265,266], 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.
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., [295,318]), and numerical software simulation is described in .
The force on the th mass can be expressed as
Following the classical derivation of the stiff-string wave equation [317,144], an obvious way to introduce stiffness in the mass-spring chain is to use a bundle of mass-spring chains to form a kind of ``lumped stranded cable''. One section of such a model is shown in Fig.9.27. Each mass is now modeled as a 2D mass disk. Complicated rotational dynamics can be avoided by assuming no torsional waves (no ``twisting'' motion) (§B.4.20).
A three-spring-per-mass model is shown in Fig.9.28 . The spring positions alternate between angles , say, on one side of a mass disk and on the other side in order to provide effectively six spring-connection points around the mass disk for only three connecting springs per section. This improves isotropy of the string model with respect to bending direction.
A problem with the simple mass-spring-chain-bundle is that there is no resistance whatsoever to shear deformation, as is clear from Fig.9.29. To rectify this problem (which does not arise due implicit assumptions when classically deriving the stiff-string wave equation), diagonal springs can be added to the model, as shown in Fig..
or, in vector form,
Digitizing via the centered second-order difference [Eq.(7.5)]
Note that requiring three adjacent spatial string samples to be in contact with the piano hammer during the attack (which helps to suppress aliasing of spatial frequencies on the string during the attack) implies a sampling rate in the vicinity of 6 megahertz . Thus, the model is expensive to compute! However, results to date show a high degree of accuracy, as desired. In particular, the stretching of the partial overtones in the stiff-string model of Fig. has been measured to be highly accurate despite using only three spring attachment points on one side of each mass disk .
Commuted Piano Synthesis
Nonlinear Piano Strings