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
![$\displaystyle f(t) = m a(t) \isdef m \dot v(t) \isdef m \ddot x(t).
$](http://www.dsprelated.com/josimages_new/pasp/img1576.png)
Since impedance is defined in terms of force and velocity, we will prefer the
form
. By the differentiation theorem for Laplace transforms
[284],8.1we have
![$\displaystyle F(s) = m [s V(s) - v(0)].
$](http://www.dsprelated.com/josimages_new/pasp/img1578.png)
![$\displaystyle F(s) = m s V(s),
$](http://www.dsprelated.com/josimages_new/pasp/img1579.png)
![$ F(s)/V(s)$](http://www.dsprelated.com/josimages_new/pasp/img1580.png)
![$\displaystyle R_m(s) \isdef m s.
$](http://www.dsprelated.com/josimages_new/pasp/img1581.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
![$\displaystyle \Gamma_m(s) \isdef \frac{1}{ms}
$](http://www.dsprelated.com/josimages_new/pasp/img1582.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
Since we normally think of an applied force as an input and the resulting
velocity as an output, the corresponding transfer function is
. The system diagram for this view
is shown in Fig. 7.3.
The impulse response of a mass, for a force input and velocity output, is defined as the inverse Laplace transform of the transfer function:
![$\displaystyle \gamma_m(t) \isdef {\cal L}^{-1}\left\{\Gamma_m(s)\right\} = \frac{1}{m}u(t)
$](http://www.dsprelated.com/josimages_new/pasp/img1584.png)
![$ \delta(t)$](http://www.dsprelated.com/josimages_new/pasp/img1585.png)
![$ m$](http://www.dsprelated.com/josimages_new/pasp/img6.png)
![$ v(t)$](http://www.dsprelated.com/josimages_new/pasp/img61.png)
![$ v(t)=1/m$](http://www.dsprelated.com/josimages_new/pasp/img1586.png)
![]() |
Once the input and output signal are defined, a transfer function is
defined, and therefore a frequency response is defined [485].
The frequency response is given by the transfer function evaluated on
the axis in the
plane, i.e., for
. For the ideal mass,
the force-to-velocity frequency response is
![$ -6$](http://www.dsprelated.com/josimages_new/pasp/img1589.png)
![$ -\pi /2$](http://www.dsprelated.com/josimages_new/pasp/img75.png)
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
.
Next Section:
Ideal Spring
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Dashpot