Lumped Models

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*
[321])
work perfectly well, albeit at a higher computational expense for a
given model quality [69,146]. 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

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University