### 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 :

*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:#### Finite Difference Implementation

Digitizing via the centered second-order difference [Eq.(7.5)]*explicit finite-difference scheme*(§D.1):

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Nonlinear Piano Strings