### Ideal Mass

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
[284],^{8.1}we have

*integrator*. Thus, an ideal mass integrates the applied force (divided by ) to produce the output velocity. This is just a ``linear systems'' way of saying force equals mass times acceleration.

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:

*unit momentum*to the mass at time 0. (Recall that momentum is the integral of force with respect to time.) Since momentum is also equal to mass times its velocity , it is clear that the unit-momentum velocity must be .

Once the input and output signal are defined, a transfer function is
defined, and therefore a *frequency response* is defined [485].
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

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
.

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Ideal Spring

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Dashpot