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
[286],^{8.1}we have

If we assume the initial velocity of the mass is zero, we have

and the impedance of the mass in the frequency domain is simply

The admittance of a mass is therefore

This is the transfer function of an

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 corresponds to transferring a

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 dB per octave, and the phase shift is radians at all frequencies.

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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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University