Introduction to Lumped Models

This appendix introduces modeling of ``lumped'' physical systems, such as configurations of masses, springs, and ``dashpots''.

The term ``lumped'' comes from electrical engineering, and refers to lumped-parameter analysis, as opposed to distributed-parameter analysis. Examples of ``distributed'' systems in musical acoustics include ideal strings, acoustic tubes, and anything that propagates waves. In general, a lumped-parameter approach is appropriate when the physical object has dimensions that are small relative to the wavelength of vibration. Examples from musical acoustics include brass-players' lips (modeled using one or two masses attached to springs--see Chapter 8), and the piano hammer (modeled using a mass and nonlinear spring, as discussed in Chapter 5). In contrast to these lumped-modeling examples, the vibrating string is most efficiently modeled as a sampled distributed-parameter system, as discussed in Chapter 4, although lumped models of strings (using, e.g., a mass-spring-chain [298]) work perfectly well, albeit at a higher computational expense for a given model quality [66,135]. In the realm of electromagnetism, distributed-parameter systems include electric transmission lines and optical waveguides, while the typical lumped-parameter systems are ordinary RLC circuits (connecting resistors, capacitors, and inductors). Again, whenever the oscillation wavelength is large relative to the geometry of the physical component, a lumped approximation may be considered. As a result, there is normally a high-frequency limit on the validity of a lumped-parameter model. For the same reason, there is normally an upper limit on physical size as well.

We begin with the fundamental concept of impedance, and discuss the elementary lumped impedances associated with springs, mass, and dashpots. These physical objects are analogous to capacitors, inductors, and resistors in lumped electrical circuits. Next, we discuss general interconnections of such elements, characterized at a single input/output location by means of one-port network theory. In particular, we will see that all passive networks present a positive real impedance at any port (input/output point). A network diagram may be replaced by an impedance diagram, which may then be translated into its equivalent circuit (replacing springs by capacitors, masses by inductors, and dashpots by resistors).

Next, we discuss digitization of lumped networks by various means, including finite differences, the bilinear transformation, and a few other simple methods for digitizing an equivalent circuit one element at a time.

In Appendix N, the subject of wave digital filters is introduced, followed by Appendix P on transfer-function modeling, which involves formulating model digitization as a digital filter design problem.