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

*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
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].

The force on the th mass can be expressed as

where

#### A Stiff Mass-Spring String Model

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
[266]. 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..

In the simulation results reported in [266], the spring-constants of the shear springs were chosen so that their stiffness in the longitudinal direction would equal that of the longitudinal springs.

#### Nonlinear Piano-String Equations of Motion in State-Space Form

For the flexible (non-stiff) mass-spring string, referring to Fig.9.26 and Eq.(9.34), we have the following equations of motion:

or, in vector form,

#### Finite Difference Implementation

Digitizing via the centered second-order difference [Eq.(7.5)]

*explicit finite-difference scheme*(§D.1):

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 [265]. 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 [265].

See [53] for alternative finite-difference formulations that better preserve physical energy and have other nice properties worth considering.

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Commuted Piano Synthesis

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Nonlinear Piano Strings