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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About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.