#### 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