The concept of impedance is central in classical electrical
engineering. The simplest case is Ohm's Law for a resistor
:
where
where
Thanks to the Laplace transform [452]2.15(or Fourier transform [454]),
the concept of impedance easily extends to masses and springs as well.
We need only allow impedances to be frequency-dependent. For
example, the Laplace transform of Newton's
yields, using the
differentiation theorem for Laplace transforms [452],
where
Specializing the Laplace transform to the Fourier transform by setting
Similarly, the impedance of a spring having spring-constant
The important benefit of this frequency-domain formulation of impedance is that it allows every interconnection of masses, springs, and dashpots (every RLC equivalent circuit) to be treated as a simple resistor network, parametrized by frequency.
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As an example, Fig.1.11 gives the impedance diagram corresponding to the equivalent circuit in Fig.1.10. Viewing the circuit as a (frequency-dependent) resistor network, it is easy to write down, say, the Laplace transform of the force across the spring using the voltage divider formula:
These sorts of equivalent-circuit and impedance-network models of mechanical systems, and their digitization to digital-filter form, are discussed further in Chapter 7.