Lumped Mass-Spring Networks

The use of a lumped network, generally of mechanical elements such as masses and springs, as a musical sound synthesis construct, is an intuitively appealing one. It was proposed by Cadoz, Luciani and Florens in the early 1980s [38], and became the basis for the CORDIS, and CORDIS-ANIMA synthesis environments [87,39,214]; it constituted the first large scale attempt at physical modeling sound synthesis. It is also the technique which is most similar to the direct simulation approaches discussed throughout the remainder of this book, though the emphasis here is entirely on fully distributed modelling, rather than lumped representations.

The framework is very simply described in terms of interactions among lumped masses, connected by springs and damping elements; when Newton's laws are employed to describe the inertial behaviour of the masses, the dynamics of such a system may be described by a set of ordinary differential equations. Interaction may be introduced through so-called ``conditional links," which can represent nonlinear contact forces. Time integration strategies, similar to those introduced in Chapter 3 in this book, operating at the audio sample rate (or sometimes above, in order to reduce frequency warping effects),in order to generate sound output. The basic operation of this method will be described in §3.4.

Figure 1.6: Lumped mass-spring networks: (a) In a linear configuration corresponding to a model of a lossless string, (b) in a 2D configuration corresponding to a model of a lossless membrane, and (c) an unstructured network, without a distributed interpretation.

Clearly, with a large enough collection of interconnected masses, a distributed object such as a string, as shown in Figure 1.6(a), or membrane, as shown in Figure 1.6(b), may be modelled. See Figure 1.6. Such configurations will be discussed explicitly in §6.4.1 and §10.1.3, respectively. A rather large philosophical distinction between the CORDIS framework and that described here is that one can develop lumped networks which are, in a sense, only quasi-physical, in that they do not correspond to recognizable physical objects, though the physical underpinnings of Newton's Laws remain. See Figure 1.6(c). Accurate simulation of complex distributed systems has not been a major concern of the designers of the CORDIS; rather, the interest is in user issues such as the modularity of lumped network structures, and the ability to interact through external control. In short, it is perhaps best to think of CORDIS as a system designed for artists and composers, rather than scientists.