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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 §9.7), and the piano hammer (modeled using a
mass and nonlinear spring, as discussed in §9.4). In
contrast to these lumped-modeling examples, the vibrating string is
most efficiently modeled as a sampled distributed-parameter system, as
discussed in
Chapter 6, although lumped models of strings (using, *e.g.*,
a *mass-spring-chain*
[318])
work perfectly well, albeit at a higher computational expense for a
given model quality [69,145]. 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 lumped-parameter 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).

In the following chapter, we discuss *digitization* of lumped
networks by various means, including *finite differences* and the
*bilinear transformation*.

- Impedance

- One-Port Network Theory
- Series Combination of One-Ports
- Mass-Spring-Wall System
- Parallel Combination of One-Ports
- Spring-Mass System
- Mechanical Impedance Analysis
- General One-Ports
- Passive One-Ports

- Digitization of Lumped Models
- Finite Difference Approximation
- Bilinear Transformation
- Application of the Bilinear Transform
- Limitations of Lumped Element Digitization

- More General Finite-Difference Methods
- General Nonlinear ODE
- Forward Euler Method
- Backward Euler Method
- Trapezoidal Rule
- Newton's Method of Nonlinear Minimization
- Semi-Implicit Methods
- Semi-Implicit Backward Euler
- Semi-Implicit Trapezoidal Rule
- Summary
- Further Reading in Nonlinear Methods

- Summary of Lumped Modeling

Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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