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A Stiff Mass-Spring String Model

Following the classical derivation of the stiff-string wave equation [320,145], 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).

Figure 9.27: Adding bending stiffness to the mass-spring string model.
\includegraphics{eps/massspringsmass}

A three-spring-per-mass model is shown in Fig.9.28 [268]. The spring positions alternate between angles $ (0,2\pi/3,4\pi/3)$ , say, on one side of a mass disk and $ (\pi/3,\pi,5\pi/3)$ 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.

Figure 9.28: Stiff mass-spring chain with alternating three-spring placement.
\includegraphics[width=0.8\twidth]{eps/masssprings3circ}

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.[*].

Figure 9.29: Illustration of the need for shear stiffness in the model.
\includegraphics{eps/massspringsshearedai}

Figure: Geometry of added shear springs.
\includegraphics[width=0.4\twidth]{eps/masssprings3shear}
In the simulation results reported in [268], 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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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-06-11 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA