The concept of impedance is central in classical electrical engineering. The simplest case is Ohm's Law for a resistor :
where denotes the voltage across the resistor at time , and is the current through the resistor. For the corresponding mechanical element, the dashpot, Ohm's law becomes
where is the force across the dashpot at time , and is its compression velocity. The dashpot value is thus a mechanical resistance.2.14
Thanks to the Laplace transform 2.15(or Fourier transform ), 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 ,
where denotes the Laplace transform of (`` ''), and similarly for displacement, velocity, and acceleration Laplace transforms. (It is assumed that all initial conditions are zero, i.e., .) The mass impedance is therefore
Specializing the Laplace transform to the Fourier transform by setting gives
Similarly, the impedance of a spring having spring-constant is given by
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.
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.