#### 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.

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

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Checking the Approximations