Nonlinear Plates

Perhaps the most dramatic example of nonlinear behaviour in all of musical acoustics is afforded by the vibration of a metal plate at high amplitudes. Various percussion instruments, and especially gongs and cymbals, exhibit this behaviour. The perceptual results are extremely strong, including effects such as the rapid buildup of high-frequency energy as heard in cymbal crashes, subharmonic generation, as well as pitch glides, which were discussed in some detail in the case of the string in Chapter 7. For sound synthesis purposes, a linear model is wholly insufficient.

There are a variety of models of nonlinear plate vibration; when the plate is thin, and vibration amplitudes are low, all of these reduce to the Kirchhoff model discussed in §10.2. (Plates which appear in a musical setting are generally thin, and so there is little reason to delve into the much more involved topic of thick plate vibration, which, even in the linear case, is orders of magnitude more involved than dealing with simple thin plate models.) Perhaps the simplest nonlinear thin plate model is that of Berger [19], which is discussed in §11.1; this system is a 2D analogue of the Kirchhoff-Carrier, or ``tension-modulated" string discussed in §8.1, and the predominant perceptual result of employing such a model is the simulation of pitch glides. Though this model is sufficient in the first instance, and leads to computationally attractive finite difference schemes, it is not capable of rendering the more interesting effects mentioned above, which are defining characteristics of some percussion instruments. To this end, the more complex model of von Karman [151,213] is introduced in §11.2, as are robust finite difference schemes based on energy conservation properties. Applications to cymbal and gong vibration appear in §11.3.

References for this chapter include: [100,170,16,138,213,151,19,128,247,48,221,217,194,73,127,128,24,25,218]