The concept of impedance extends also to masses and springs.
Figure 7.2 illustrates an ideal mass of
kilograms
sliding on a frictionless surface. From Newton's second law of motion, we
know force equals mass times acceleration, or
Since impedance is defined in terms of force and velocity, we will prefer the
form
. By the differentiation theorem for Laplace transforms
[286],8.1we have
If we assume the initial velocity of the mass is zero, we have
and the impedance
The admittance of a mass
This is the transfer function of an integrator. Thus, an ideal mass integrates the applied force (divided by
Since we normally think of an applied force as an input and the resulting
velocity as an output, the corresponding transfer function is
. The system diagram for this view
is shown in Fig. 7.3.
The impulse response of a mass, for a force input and velocity output, is defined as the inverse Laplace transform of the transfer function:
In this instance, setting the input to
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Once the input and output signal are defined, a transfer function is
defined, and therefore a frequency response is defined [488].
The frequency response is given by the transfer function evaluated on
the
axis in the
plane, i.e., for
. For the ideal mass,
the force-to-velocity frequency response is
Again, this is just the frequency response of an integrator, and we can say that the amplitude response rolls off
In circuit theory, the element analogous to the mass is the inductor,
characterized by
, or
. In an analog
equivalent circuit, a mass can be represented using an inductor with value
.