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 [
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
frequencydependent. For
example, the Laplace transform of Newton's
yields, using the
differentiation theorem for Laplace transforms [
449],
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 springconstant
is given by
The important benefit of this
frequencydomain 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 forcedriven, 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 (frequencydependent) 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 equivalentcircuit and impedancenetwork models of
mechanical systems, and their digitization to
digitalfilter form, are
discussed further in Chapter
7.
Next Section: Wave Digital FiltersPrevious Section: Equivalent Circuits