## Summary of Lumped Modeling

In this chapter, we looked at the fundamentals of lumped modeling elements such as masses, springs, and dashpots. The important concept of driving-point impedance was defined and discussed, and electrical equivalent circuits were developed, along with associated elementary (circuit) network theory. Finally, we looked at basic ways of digitizing lumped elements and more complex ODEs and PDEs, including a first glance at some nonlinear methods.

Practical examples of lumped models begin in §9.3.1. In
particular, piano-like models require a ``hammer'' to strike the
string, and §9.3.1 explicates the simplest case of an
ideal point-mass striking an ideal vibrating string. In that model,
when the mass is in contact with the string, it creates a
*scattering junction* on the string having reflection and
transmission coefficients that are first-order filters. These filters
are then digitized via the bilinear transform. The ideal string
itself is of course modeled as a digital waveguide. A detailed
development of wave scattering at impedance-discontinuities is
presented for digital waveguide models in §C.8, and for wave
digital filters in Appendix F.

#### Outline

In this chapter, we will look at a variety of ways to digitize macroscopic point-to-point transfer functions corresponding to a desired impulse response :

- Sampling to get
- Pole mappings (such as followed by Prony's method)
- Modal expansion
- Frequency-response matching using digital filter design methods

Next, we'll look at the more specialized technique known as
*commuted synthesis*, in which computational efficiency may be
greatly increased by interchanging (``commuting'') the series order of
component transfer functions. Commuted synthesis delivers large gains
in efficiency for systems with a short excitation and high-order
resonators, such plucked and struck strings. In Chapter 9,
commuted synthesis is applied to piano modeling.

Extracting the least-damped modes of a transfer function for separate parametric implementation is often used in commuted synthesis. We look at a number of ways to accomplish this goal toward the end of this chapter.

We close the chapter with a simple example of transfer-function
modeling applied to the digital *phase shifter* audio effect.
This example classifies as *virtual analog* modeling, in which a
valued analog device is converted to digital form in a way that
preserves all valued features of the original. Further examples of
transfer-function models appear in Chapter 9.

**Next Section:**

Sampling the Impulse Response

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More General Finite-Difference Methods