The concept of impedance is central in classical electrical
engineering. The simplest case is Ohm's Law for a resistor
denotes the voltage across the resistor at time
is the current through the resistor. For the corresponding mechanical
element, the dashpot
's law becomes
is the force
across the dashpot
is its compression velocity
. The dashpot value
is thus a
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 ,
denotes the Laplace transform of
''), and similarly for displacement
, velocity, and
acceleration Laplace transforms. (It is assumed that all initial
are zero, i.e.
The mass impedance
Specializing the Laplace transform to the Fourier transform by setting
Similarly, the impedance of a spring
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.
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:
These sorts of equivalent-circuit and impedance-network models of
mechanical systems, and their digitization to digital-filter
discussed further in Chapter 7
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