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 [326]) work perfectly well, albeit at a higher computational expense for a given model quality [71,145]. 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 lumpe