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Impedance Networks

The concept of impedance is central in classical electrical engineering. The simplest case is Ohm's Law for a resistor $ R$:

$\displaystyle V(t) \eqsp R\, I(t)

where $ V(t)$ denotes the voltage across the resistor at time $ t$, and $ I(t)$ is the current through the resistor. For the corresponding mechanical element, the dashpot, Ohm's law becomes

$\displaystyle f(t) \eqsp \mu\, v(t)

where $ f(t)$ is the force across the dashpot at time $ t$, and $ v(t)$ is its compression velocity. The dashpot value $ \mu $ is thus a mechanical resistance.2.14

Thanks to the Laplace transform [449]2.15(or Fourier transform [451]), 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 $ f=ma$ yields, using the differentiation theorem for Laplace transforms [449],

$\displaystyle F(s) \eqsp m\, A(s) \eqsp m\, sV(s) \eqsp m\, s^2X(s)

where $ F(s)$ denotes the Laplace transform of $ f(t)$ (`` $ F(s) =
{\cal L}_s\{f\}$''), and similarly for displacement, velocity, and acceleration Laplace transforms. (It is assumed that all initial conditions are zero, i.e., $ f(0)=x(0)=v(0)=a(0)=0$.) The mass impedance is therefore

$\displaystyle R_m(s) \isdefs \frac{F(s)}{V(s)} \eqsp ms.

Specializing the Laplace transform to the Fourier transform by setting $ s=j\omega$ gives

$\displaystyle R_m(j\omega) \eqsp jm\omega.

Similarly, the impedance of a spring having spring-constant $ k$ is given by

R_k(s) &=& \frac{k}{s}\\ [5pt]
R_k(j\omega) &=& \frac{k}{j\omega}.

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.

Figure: Impedance diagram for the force-driven, series arrangement of mass and spring shown in Fig.1.9.

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:

$\displaystyle F_k(s) \eqsp F_{\mbox{ext}}(s) \frac{R_k(s)}{R_m(s)+R_k(s)} \eqsp F_{\mbox{ext}}(s)\frac{k/m}{s^2+k/m}

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.

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