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The Oscillator in Musical Acoustics

Finite difference schemes for the simple harmonic oscillator were introduced in some detail in the last chapter. Though the SHO is, by itself, not an extremely interesting system, many excitation mechanisms in musical instruments may be modeled as variants of it, generally nonlinear. Some examples which will be discussed in this chapter are the hammer/mallet interaction, reeds (including lip models), the bow, and the glottis.

When designing a sound synthesis simulation, particularly for nonlinear systems, there are different ways of proceeding. It is sometimes useful, in the first instance, to make use of an ad hoc numerical method, generally efficient and easily programmed, in order to generate sound output quickly, perhaps so as to uncover problems in the formulation of the underlying model system. On the other hand, with some extra work (and generally at some additional computational expense), better behaved numerical methods may be derived. Robustness, or numerical stability under a wide variety of possible playing conditions, is an especially useful property for a synthesis algorithm to possess, particularly if the physical structure it simulates is to be virtually connected to other such structures.

Frequency domain analysis may indeed be extended to deal with nonlinear systems such as those discussed in this chapter, through perturbation techniques such as Linstedt-Poincaré methods, or harmonic balance techniques [151]. The conclusions one may reach in this manner are generally extremely complex, and are not particularly useful when it comes to the analysis of numerical stability for associated numerical methods. Energy techniques will thus be employed whenever possible. It is worth noting that through the use of such techniques, though generally applicable to all the nonlinear systems discussed in this chapter, simple results are obtained only when nonlinearities of a smooth type, i.e., those which may be expressed in terms of polynomial terms. See §4.1. Though in the case of excitation mechanisms, the nonlinearities here are not of this form, some time will be spent on energy methods, mainly because which the nonlinearities which arise in distributed problems, such as strings and plates, generally are quite smooth. Simple ad hoc methods for the variety of excitation mechanisms mentioned above will also be presented in §4.2 to §4.6, and general collision models will also be discussed in §4.7.

References for this chapter include: [125,45,212,245,123,182,183,56,8,168,196]



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next up previous contents index
Next: Nonlinear Oscillators of Series Up: Numerical Sound Synthesis Previous: Programming Exercises   Contents   Index
Stefan Bilbao 2006-11-15